finished example DecomposeOnceByRandomEndomorphism and solved the doc…

…umentation problem.

doc/Doc.autodoc

@@ -6,7 +6,7 @@
 @InsertChunk CategoryOfRepresentations
 
 @Subsection One step in the direct sum decomposition
-@InsertChunk DecomposeOnceByRandomEndomorphism
+@InsertChunk DecomposeOnce
 
 @Subsection Another category of module homomorphisms
 @InsertChunk RepresentingC4C4

examples/DecomposeOnceByRandomEndomorphism.g

@@ -1,8 +1,8 @@
-#! @BeginChunk DecomposeOnceByRandomEndomorphism
+#! @BeginChunk DecomposeOnce
 
 LoadPackage( "CatReps" );
 
-#! @Example
+#! @BeginExample
 c3c3 := ConcreteCategoryForCAP( [ [2,3,1], [4,5,6], [,,,5,6,4] ] );
 #! A finite concrete category
 GF3 := HomalgRingOfIntegers( 3 );
@@ -22,11 +22,14 @@ DecomposeOnceByRandomEndomorphism( nine );
 #! fail
 #! @EndExample
 
-#! The above shows that our representation <C>nine</C> is indecomposable.
+#! The above shows that our representation <C>nine</C> is 
+#! indecomposable (with a high probability).
 #! We use the tensor product to generate another representation
-#! <C>fortyone</C>, that is hopefully decomposable.
+#! <C>fortyone</C>, that is hopefully decomposable, and
+#! inspect the two resulting embeddings <C>iota</C> and
+#! <C>kappa</C>.
 
-#! @Example
+#! @BeginExample
 fortyone := TensorProductOnObjects( nine, nine );
 #! <(1)->25, (2)->16; (a)->25x25, (b)->25x16, (c)->16x16>
 result := DecomposeOnceByRandomEndomorphism( fortyone );
@@ -227,12 +230,13 @@ Display( S );
 #! matrix on the diagonal: A $3\times 3$-matrix from <C>S(kq.a)</C>,
 #! a $3 \times 1$-matrix from <C>S(kq.b)</C> and a $1\times 1$-matrix
 #! from <C>S(kq.c)</C>. This matches with the source of the 
-#! natural transformation $\iota$.
+#! embedding <C>iota</C>.
 
-#! @Example
+#! @BeginExample
 Display( iota );
-#! A morphism in The category of functors: Algebroid generated by the right quiver q(2)[a:1->1,b:1->2,c:\
-#! 2->2] -> Category of matrices over GF(3) defined by the following data:
+#! A morphism in The category of functors: Algebroid generated by the
+#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices
+#! over GF(3) defined by the following data:
 #! 
 #! 
 #! Image of <(1)>:
@@ -280,12 +284,12 @@ Display( Source( iota) );
 #! A morphism in Category of matrices over GF(3)
 #! @EndExample
 
-#! We can then look at the other factor of the direct sum 
-#! decomposition, i.e. $\kappa$. The iteration of 
+#! We can then look at the other embedding of the direct sum 
+#! decomposition, <C>kappa</C>. The iteration of 
 #! <C>WeakDirectSumDecomposition</C> will continue then
 #! with <C>Source( kappa )</C>. Each time the random
-#! endomorphism will decompose the representation by
-#! at most a dimension of $3$.
+#! endomorphism will decompose the representation,
+#! lowering the dimensions of each object at most by $3$.
 
 #! @Example
 Source( kappa );